Exam 4: Exponential and Logarithmic Functions

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The range of an exponential function $f ( x ) = b ^ { x }$ is .

Choose the one alternative that best completes the statement or answers the question. Evaluate the function at the given value of x. Round to 4 decimal places if necessary. - $f ( x ) = \left( \frac { 1 } { 7 } \right) ^ { x } ; \quad f ( 0.4 e )$

Find an equation for the inverse function. - $f ( x ) = 10 ^ { x + 1 } + 5$

Determine if the statement is true or false. - $\log \left( \frac { x } { y } \right) = \frac { \log x } { \log y }$

Solve the equation. - $\frac { e ^ { x } - 19 \cdot e ^ { - x } } { 3 } = 6$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The equivalence property of logarithmic expressions indicates that if $\log _ { b } x = \log _ { b } y$ , then = .

Use the change-of-base and a calculator to approximate the logarithm to 4 decimal places. - $\log _ { 5 } 9$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The product property of logarithms indicates that $\log _ { b } ( x y ) =$ ـــــــــــــfor positive real numbers $b , x$ , and $y$ where $b \neq 1$ .

Solve the problem. -Use the graph of $y = \left( \frac { 1 } { 3 } \right) ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x + 1 } + 3$

Solve the problem. -Use transformations of the graph of $y = \log _ { 4 } x$ to graph the function. $y = \log _ { 4 } ( x - 3 ) - 4$

Solve the problem. -The millage rate is the amount of property tax per $1000 of the taxable value of a home. For a certain county the millage rate is 29 mil. A city within the county also imposes a flat fee of $101 per Home. a. Write a function representing the total amount of property tax $T ( x )$ for a home with a taxable value thousand dollars. b. Write an equation for $T ^ { - 1 } ( x )$ . c. What does the inverse function represent in the context of this problem?

Solve the problem. -The atmospheric pressure on an object decreases as altitude increases. If a is the height (in km) above sea level, then the pressure P(a) (in mmHg) is approximated by P(a) = 760e-0.13a. Determine The atmospheric pressure at 7.176 km. Round to the nearest whole unit.

Write the word or phrase that best completes each statement or answers the question. - $\log _ { 3 } \frac { 1 } { 729 } = - 6$

Choose the one alternative that best completes the statement or answers the question. Solve for the indicated variable. - $\frac { 1 } { k } \ln \left( \frac { P } { 16.6 } \right) = A$ for $P$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The formula $A = P e ^ { r t }$ gives the amount A in an account after t years at an interest rate r under the assumption that interest is compounded .

Solve the problem. -The monthly costs for a small company to do business has been increasing over time in part due to inflation. The table gives the monthly cost y (in $) for the month of January for selected years. The Variable t represents the number of years since 2008. $\begin{array}{c|c}\begin{array}{c}\text { Year } \\(t=\mathbf{0} \text { is 2008) }\end{array} & \begin{array}{c}\text { Monthly } \\\text { Costs (\$) } y\end{array} \\\hline 0 & 14,000 \\1 & 14,200 \\2 & 14,400 \\3 & 14,600\end{array}$ Monthly Cost (Jan.) for Selected Years Year $( \mathrm { t } = 0$ represents $2008 )$ a. Use a graphing utility to find a model of the form $y = a b ^ { t }$ . b. Write the function from part (a) as an exponential function with base $e$ . c. Use the model to predict the monthly cost for January in the year 2,017 if this trend continues. Round to the nearest hundred dollars.

Solve the equation. - $\log _ { 5 } y + \log _ { 5 } ( 12 - y ) = \log _ { 5 } 35$

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - $4 \log _ { 2 } ( 9 p - 95 ) = 8$

Write as the sum or difference of logarithms and fully simplify, if possible. Assume the variable represents a positive real number. - $\ln \left( \frac { \sqrt [ 5 ] { a b } } { c ^ { 4 } d } \right)$

Given the function f : = {(1, 2), (2, 3), (3, 4)} write the set of ordered pairs representing $f ^ { - 1 }$